1 answer

The result of t[e_uv,:] cannot be the same as t[:] with a mere substitution x=(u+v)/2 and y=(u-v)/2 because it gives the components of t in a different vector frame: the frame is e_uv for t[e_uv,:], whereas it is the default frame on M for t[:], which is e_xy. In other words, the components returned by t[e_uv,:] are those of the following expansion:

Comments

Thanks Eric, I understand now.
As you said, it is not a mere substitution, it is also a change of basis.
For a tensor T you have to multiply by $J^{-1}\cdot T\cdot J$
For a vector V you have to multiply by $J^{-1}\cdot V$
Where J is the Jacobian of uv_to_xy against u,v
Thanks again.
Daniel